Ground state energy threshold and blow-up for NLS with competing nonlinearities

TL;DR

This paper investigates the ground state energy threshold in nonlinear Schrödinger equations with competing nonlinearities, establishing conditions for its achievement and linking it to finite-time blow-up solutions.

Contribution

It provides a complete characterization of the ground state energy achievement and blow-up solutions for NLS with combined nonlinearities, depending on the mass and type of perturbation.

Findings

Ground state energy is achieved by radially symmetric solutions for certain perturbations.

Existence of a critical mass for achieving ground state energy in subcritical cases.

Finite-time blow-up solutions exist below the ground state energy threshold.

Abstract

We consider the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover, the ground state energy is not achieved for mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite time, for any energy below the ground state energy threshold.